aa r X i v : . [ m a t h . F A ] D ec A DIRAC DELTA OPERATOR
JUAN CARLOS FERRANDO
Abstract. If T is a (densely defined) self-adjoint operator acting on a complex Hilbert space H and I stands for theidentity operator, we introduce the delta function operator λ δ ( λI − T ) at T . When T is a bounded operator, then δ ( λI − T ) is an operator-valued distribution. If T is unbounded, δ ( λI − T ) is a more general object that still retainssome properties of distributions. We derive various operative formulas involving δ ( λI − T ) and give several applicationsof its usage. The delta function δ ( λI − T )The scalar delta ‘function’ λ δ ( λ − a ) along with itsderivatives were introduced by Paul Dirac in [1], and laterin [2, Section 15], although its definition can be tracedback to Heaviside. The rigorous treatment of this objectin the context of distribution theory is due to LaurentSchwartz [6, 12]. In this paper we extend the definition of δ ( λ − a ) from real numbers to self-adjoint operators on aHilbert space H . We denote by D ( R ) = lim −→ D ([ − n, n ])the linear space of infinitely differentiable complex-valuedfunctions of compact support, equipped with the inductivelimit topology. As usual in physics we shall assume thatthe scalar product in H is anti-linear for the first variable.If T is a densely defined self-adjoint operator on H and I stands for the identity operator, we define the deltafunction operator λ δ ( λI − T ) at T by(1.1) f ( T ) = Z + ∞−∞ f ( λ ) δ ( λI − T ) dλ for each f ∈ C ( R ), i. e., for each real-valued continuousfunction f ( λ ). Here dλ is the Lebesgue measure of R , butthe right-hand side of (1.1) is not a true integral. If T isa bounded operator, we shall see at once that δ ( λI − T )must be regarded as a vector-valued distribution , i. e., asa continuous linear map from the space D ( R ) into thelocally convex space L ( H ) of the bounded linear operators(endomorphisms) on H equipped with the strong operatortopology [10, 11], whose action on f ∈ D ( R ) we denote asan integral. If T is unbounded we shall see that δ ( λI − T )still retains some useful distributional-like properties. Theprevious equation means(1.2) h y, f ( T ) x i = Z + ∞−∞ f ( λ ) h y, δ ( λI − T ) x i dλ for each ( x, y ) ∈ D ( f ( T )) × H , where D ( f ( T )) stands forthe domain of the self-adjoint operator f ( T ).Let us recall that if T is a (densely defined) self-adjointoperator, there is a unique spectral family { E λ : λ ∈ R } In what follows σ ( T ) will denote the spectrum of T . Recall thatthe residual spectrum of a self-adjoint operator T is empty, so that σ ( T ) = σ p ( T ) ∪ σ c ( T ), where σ p ( T ) denotes the point spectrum (the eigenvalues) and σ c ( T ) the continuous spectrum of T . of self-adjoint operators defined on the whole of H thatsatisfy ( i ) E λ ≤ E µ and E λ E µ = E λ for λ ≤ µ , ( ii )lim ǫ → + E λ + ǫ x = E λ x , and ( iii ) lim λ →−∞ E λ x = andlim λ →∞ E λ x = x in H for all x ∈ H . The domain D ( T )of T consists of those x ∈ H such that Z + ∞−∞ | λ | d k E λ x k < ∞ . In this case, the spectral theorem ( cf . [8, Section 107]) andthe Borel-measurable functional calculus provide a self-adjoint operator f ( T ) defined by(1.3) f ( T ) = Z + ∞−∞ f ( λ ) dE λ for each Borel-measurable function f ( λ ), whose domain D ( f ( T )) = (cid:26) x ∈ H : Z + ∞−∞ | f ( λ ) | d k E λ x k < ∞ (cid:27) is dense in H . Observe that if T is bounded, f ( T ) neednot be bounded. Moreover, since λ E λ is constant onthe set R \ σ ( T ) of T , an open set in R , equation (1.3) tellus that f ( λ ) need not be defined on R \ σ ( T ).Thanks to (1.3) the definition of δ ( λI − T ) may be ex-tended to Borel-measurable functions by declaring thatthe equation (1.1) holds for ( x, y ) ∈ D ( f ( T )) × H andeach Borel function f . But, by reasons that will becomeclear later, we shall restrict ourselves to those Borel func-tions which are continuous at each point of σ p ( T ). More-over, working with the real and complex parts, no dif-ficulty arises if the function f involved in the equation(1.1) is complex-valued (except that f ( T ) is no longera self-adjoint operator whenever Im f = 0). Thus, un-less otherwise stated, we shall assume that both in (1.1)and (1.3) the function f is complex-valued. Note thatthe complex Stieltjes measure d h E λ x, y i need not be dλ -continuous. In what follows we shall denote by B p ( R )the linear space over C consisting of all complex-valuedBorel-measurable functions of one real variable which arecontinuous on σ p ( T ).If f n → f in D ( R ), the sequence { f n } ∞ n =1 is uniformlybounded and f n ( x ) → f ( x ) at each x ∈ R . So, if T isbounded on H (equivalently, self-adjoint on the whole of H ) it turns out that f n ( T ) → f ( T ) in the strong opera-tor topology [3, 10.2.8 Corollary]. Therefore, in this case δ ( λI − T ) is an L ( H )-valued distribution.As all integrals considered so far are over σ ( T ), we have(1.4) δ ( λI − T ) = ∀ λ / ∈ σ ( T ) . Also δ ( − λI + T ) = δ ( λI − T ) for all λ ∈ R . On the otherhand, if µ ∈ σ p ( T ) and y is an eigenvector correspondingto the eigenvalue µ , clearly(1.5) δ ( λI − T ) y = δ ( λ − µ ) y for every λ ∈ R . In the particular case when T a is thelinear operator defined on H by T a x = ax for a fixed a ∈ R , then T a is a self-adjoint linear operator with σ ( T a ) = σ p ( T a ) = { a } . In this case δ ( λI − T a ) x = δ ( λ − a ) x forevery x ∈ H , i. e., δ ( λI − T a ) = δ ( λ − a ) I .Since equality D f ( T ) † y, x E = h y, f ( T ) x i holds for all x, y ∈ D ( f ( T )) and each f ∈ B p ( R ), we may infer that h y, δ ( λI − T ) x i = h δ ( λI − T ) y, x i holds (in a ‘distributional’ sense) for all x, y ∈ D ( T ). Thissuggests that in certain sense δ ( λI − T ) may be regarded(possibly for almost all λ ∈ R ) as a Hermitian operator on D ( T ).Let us also point out that as equation (1.1) holds forall f ∈ D ( R ), in a distributional sense we have(1.6) ddλ h y, E λ x i = h y, δ ( λI − T ) x i If λ Y ( λ − µ ) denotes the unit step function at µ ∈ R ,given by Y ( λ − µ ) = 0 if λ < µ and Y ( λ − µ ) = 1 if λ ≥ µ , since E λ = Y ( λI − T ) for each λ ∈ R , formally(1.7) dE λ /dλ = Y ′ ( λI − T ) . So, from (1.6) and (1.7) we get Y ′ ( λI − T ) = δ ( λI − T ). Proposition 1. If T is a bounded self-adjoint operator on H and f ∈ C ( R ) , then Z + ∞−∞ f ( λ ) δ ′ ( λI − T ) dλ = − f ′ ( T ) . The same equality holds if T is unbounded but f ∈ D ( R ) . If T is a self-adjoint operator and f ∈ B p ( R ), then Z + ∞−∞ | f ( λ ) | δ ( λI − T ) dλ = Z + ∞−∞ | f ( λ ) | dE λ . where the latter equality is the definition of | f ( T ) | . So,we have the following result. Proposition 2. If T is self-adjoint and f ∈ B p ( R ) , then h f ( T ) y, f ( T ) x i = Z + ∞−∞ | f ( λ ) | h y, δ ( λI − T ) x i dλ for every x, y ∈ D ( f ( T )) .Proof. We adapt a classic argument. Indeed, for every x, y ∈ D ( f ( T )) we have h f ( T ) y, f ( T ) x i = Z + ∞−∞ f ( λ ) d h f ( T ) y, E λ x i . Since E µ E λ = E µ whenever µ ≤ λ , and h E λ y, x i does notdepend on µ , by splitting the integral we get(1.8) Z + ∞−∞ f ( µ ) d h E µ y, E λ x i = Z λ −∞ f ( µ ) d h y, E µ x i , where clearly the first integral is h f ( T ) y, E λ x i . Plugging d h f ( T ) y, E λ x i into (1.8), we are done. (cid:3) Corollary 3.
Under the same conditions of the previoustheorem, the equality (1.9) k f ( T ) x k = Z + ∞−∞ | f ( λ ) | h x, δ ( λI − T ) x i dλ holds for every x ∈ D ( f ( T )) . Proposition 4. If T is self-adjoint and { f n } ∞ n =1 is a uni-formly bounded sequence in B p ( R ) such that f n → f point-wise on R with f ∈ B p ( R ) , then f n ( T ) x → f ( T ) x forevery x ∈ D ( T ) .Proof. This is a straightforward consequence of precedingcorollary and the Lebesgue dominated convergence theo-rem. (cid:3)
This proposition holds in particular if f n → f in D ( R ).Hence, even in the unbounded case, δ ( λI − T ) behaves asa vector-valued distribution-like object. Proposition 5.
Let ( λ, µ ) g ( λ, µ ) be a function definedon R such that g ( λ, · ) ∈ L ( R ) for every λ ∈ R and g ( · , µ ) ∈ B p ( R ) for every µ ∈ R . If the parametric integral f ( λ ) = Z + ∞−∞ g ( λ, µ ) dµ is continuous on R and makes sense if we replace λ by aself-adjoint operator T , the value of the integral Z + ∞−∞ Z + ∞−∞ g ( λ, µ ) δ ( λI − T ) dµ dλ does not depend on the integration ordering.Proof. Since g ( · , µ ) ∈ B p ( R ) for every µ ∈ R ., one has g ( T, µ ) = Z + ∞−∞ g ( λ, µ ) δ ( λI − T ) dλ, which implies f ( T ) = Z + ∞−∞ (cid:26)Z + ∞−∞ g ( λ, µ ) δ ( λI − T ) dλ (cid:27) dµ. On the other hand, by the definition of δ ( λI − T ) we have f ( T ) = Z + ∞−∞ (cid:26)Z + ∞−∞ g ( λ, µ ) dµ (cid:27) δ ( λI − T ) dλ, for ( x, y ) ∈ D ( T ) × H . So, the proposition follows. (cid:3) Theorem 6. If T is a self-adjoint operator on H , then Z + ∞ f ( λ ) δ (cid:0) λI − T (cid:1) dλ =(1.10) Z + ∞ √ λ n δ (cid:16) √ λI − T (cid:17) − δ (cid:16) √ λI + T (cid:17)o f ( λ ) dλ if λ > and f ∈ B p ( R ) , both members acting on D (cid:0) T (cid:1) . Proof.
First note that T ≥
0. Hence σ (cid:0) T (cid:1) ⊆ [0 , + ∞ ),which implies that δ (cid:0) λI − T (cid:1) = if λ <
0. Since T isa self-adjoint operator, for f ∈ B p ( R ) we have Z + ∞ f ( λ ) δ (cid:0) λI − T (cid:1) dλ = f (cid:0) T (cid:1) On the other hand, it is clear that Z + ∞ f ( λ )2 √ λ δ (cid:16) √ λI − T (cid:17) dλ = Z + ∞ f (cid:0) µ (cid:1) δ ( µI − T ) dµ whereas, using that δ ( − µI + T ) = δ ( µI − T ), we have − Z + ∞ f ( λ )2 √ λ δ (cid:16) √ λI + T (cid:17) dλ = Z −∞ f (cid:0) µ (cid:1) δ ( µI − T ) dµ So, the right-hand side of (1.10) coincides with Z + ∞−∞ f (cid:0) µ (cid:1) δ ( µI − T ) dµ = f (cid:0) T (cid:1) since µ f (cid:0) µ (cid:1) is a Borel function. (cid:3) If we denote by L ( H ) the linear space of all linear en-domorphisms on H , the next theorem summarize someprevious results. Theorem 7. If T is a densely defined self-adjoint operatoron a Hilbert space H , there is an L ( H ) -valued linear map δ T on B p ( R ) , whose action on f ∈ B p ( R ) we denote by h δ T , f i = Z + ∞−∞ f ( λ ) δ ( λI − T ) dλ, such that h δ T , f i = f ( T ) . If { f n } ⊆ B p ( R ) is uniformlybounded and f n ( t ) → f ( t ) , with f ∈ B p ( R ) , for all t ∈ R then h δ T , f n i x → h δ T , f i x for all x ∈ H . If T is bounded, δ T is an L ( H ) -valued distribution, so h δ T , f i is a boundedoperator on H . In addition δ ( λI − T ) = if λ / ∈ σ ( T ) and h y, δ ( λI − T ) x i = h δ ( λI − T ) y, x i for x, y ∈ D ( T ) . Explicit form of δ ( λI − T )If Q is a vector-valued distribution, the Fourier trans-form of Q is defined as the vector valued distribution F Q on S ( R ) such that hF Q, f i = h Q, F f i . As usual, we de-note by F − the inverse Fourier transform. Theorem 8. If T is a self-adjoint operator, the identity (2.1) δ ( λI − T ) = 12 π Z + ∞−∞ e it ( λI − T ) dt holds for every λ ∈ R , and the action f ( T ) of δ ( λI − T ) on f ∈ S ( R ) is given by f ( T ) = Z + ∞−∞ (cid:26)Z + ∞−∞ f ( λ )2 π e it ( λI − T ) dλ (cid:27) dt. Proof.
Setting δ T ( λ ) = δ ( λI − T ) observe that( F δ T ) ( t ) = 1 √ π e − itT . Indeed, if f ∈ S ( R ) we have hF δ T , f i = h δ T , F f i = Z + ∞−∞ ( F f ) ( λ ) δ ( λI − T ) dλ = ( F f ) ( T ) = 1 √ π Z + ∞−∞ f ( t ) e − itT dt. Consequently(2.2) δ T = F − (cid:26) √ π e − itT (cid:27) . Functionally, the action of δ T on f ∈ S ( R ) by means ofequation (2.2) becomes(2.3) h δ T , f i = (cid:28) √ π e − itT , (cid:0) F − f (cid:1) ( t ) (cid:29) Consequently, we have h δ T , f i = Z + ∞−∞ (cid:26)Z + ∞−∞ f ( µ )2 π e it ( µI − T ) dµ (cid:27) dt with the order of the integration as stated. (cid:3) Corollary 9. If e − itT x = x ( t ) , for x ∈ D ( T ) one has δ ( λI − T ) x = 12 π Z + ∞−∞ e iλt x ( t ) dt and if x ∈ D ( f ( T )) and f ∈ S ( R ) , then f ( T ) x = 12 π Z + ∞−∞ (cid:26)Z + ∞−∞ f ( λ ) e iλt dλ (cid:27) x ( t ) dt. Remark 10.
Consider the one-parameter unitary group { U ( t ) : t ∈ R } generated by the self-adjoint operator T ,that is, U ( t ) = exp ( − itT ) for every t ∈ R . If F denotesthe Fourier transform, equation (2.2) can be written as(2.4) δ ( λI − T ) = 1 √ π F − ( U ) ( λ ) . So, equation (2.3) reads as(2.5) f ( T ) = 1 √ π Z + ∞−∞ (cid:0) F − f (cid:1) ( t ) U ( t ) dt. In what follows we shall compute the spectral fam-ily { E λ : λ ∈ R } for some useful self-adjoint operators ofQuantum Mechanics by means of the delta δ ( λI − T ).Nonetheless, although E λ = Y ( λI − T ), the identification Y ( λI − T ) = Z + ∞−∞ Y ( λ − µ ) δ ( µI − T ) dµ might be not well-defined because µ Y ( λ − µ ) has ajump discontinuity at µ = λ . Indeed, if λ ∈ σ p ( T ) and x is an eigenvector corresponding to λ , then Z + ∞−∞ Y ( λ − µ ) δ ( µI − T ) x dµ = (Z λ −∞ δ ( µ − λ ) dµ ) x and the right-hand integral makes no sense (see [4] for auseful discussion). If λ / ∈ σ p ( T ) we define(2.6) E λ = Z + ∞−∞ Y ( λ − µ ) δ ( µI − T ) dµ If λ belongs to σ p ( T ), then ( µ Y ( λ − µ )) / ∈ B p ( R ).In order to define E λ we enlarge a little the interval ofintegration by considering the integral Z λ + ǫ −∞ δ ( µ − λ ) dµ for small ǫ >
0. So, if λ ∈ σ p ( T ) we define(2.7) E λ = lim ǫ → + Z + ∞−∞ Y ( λ + ǫ − µ ) δ ( µI − T ) dµ. The limit is well-defined since lim ǫ → + E λ + ǫ = E λ point-wise on H . In the particular case when λ belongs to σ d ( T ),the discrete part of σ p ( T ), λ is isolated in σ p ( T ). Example 11.
The spectral family of the ( up to a sign ) one-dimensional Quantum Mechanics momentum opera-tor of the free particle P = iD , where Dϕ = ϕ ′ , acting onthe Hilbert space H = L ( R ) is given by ( E λ ϕ ) ( x ) = 12 ϕ ( x ) + 12 πi p . v . Z + ∞−∞ e iλ ( s − x ) s − x ϕ ( s ) ds for every regular compactly supported ϕ ∈ D ( P ) .Proof. As is well-known P is a self-adjoint operator with D ( P ) = H , ( R ) and σ c ( P ) = R . Since (cid:0) e − itP ϕ (cid:1) ( x ) = (cid:0) e tD ϕ (cid:1) ( x ) = ϕ ( x + t )for a regular enough ϕ ∈ D ( P ), by Corollary 9 we have { δ ( µI − P ) ϕ } ( x ) = 12 π Z + ∞−∞ e iµt ϕ ( x + t ) dt. Note that the integral of the right-hand side does existbecause ϕ has compact support.According to the definition of E λ for the continuousspectrum and keeping in mind the order of integration asindicated in Corollary 9, one has { E λ ϕ } ( x ) = 12 π Z + ∞−∞ Z + ∞−∞ Y ( λ − µ ) e iµt ϕ ( x + t ) dµ dt. So, since1 √ π Z + ∞−∞ Y ( λ − µ ) e iµt dµ = F ( Y ) ( t ) · e iλt , bearing in mind the distributional relation(2.8) F ( Y ) ( t ) = r π (cid:18) δ ( t ) + 1 iπ p . v . t (cid:19) , we get { E λ ϕ } ( x ) = 12 ϕ ( x ) + 12 πi Z + ∞−∞ e iλ ( s − x ) s − x ϕ ( s ) ds where the last integral must be understood in Cauchy’sprincipal value sense. (cid:3) Example 12.
The spectral family of the one-dimensionalQuantum Mechanics kinetic energy term of the free par-ticle, corresponding to the Laplace operator T = − D on H = L ( R ) , where D ϕ = ϕ ′′ , is given by ( E λ ϕ ) ( x ) = 1 iπ p . v . Z + ∞−∞ cos ( λ ( s − x )) − s − x ϕ ( s ) ds for λ > and E λ = whenever λ < , where ϕ is aregular function with compact support belonging to D ( T ) .Proof. In this case T is a self-adjoint operator with σ ( T ) =[0 , + ∞ ). Since T = ( iD ) , according to (1.10) we have δ ( λI − T ) = 12 √ λ n δ (cid:16) √ λI − iD (cid:17) − δ (cid:16) √ λI + iD (cid:17)o regarded as a functional on S ( R ) through dλ -integrationover [0 , + ∞ ). Plugging( δ ( µI ∓ iD ) ϕ ) ( x ) = 12 π Z + ∞−∞ e iµt ϕ ( x ± t ) dt into the previous expression and keeping in mind the cor-rect order of integration, we see that Z ∞ f ( λ ) ( δ ( λI − T ) ϕ ) ( x ) dλ =14 π Z ∞ Z + ∞−∞ f ( λ ) e i √ λt √ λ [ ϕ ( x + t ) − ϕ ( x − t )] dλ dt for every f ∈ S ( R ). By the definition of E λ if λ > δ ( µI − T ) = whenever µ <
0, we have( E λ ϕ ) ( x ) = Z + ∞ Y ( λ − µ ) ( δ ( µI − T ) ϕ ) ( x ) dµ. Working out the penultimate integral with µ instead of λ and f ( µ ) = Y ( λ − µ ), we obtain Z + ∞ Z + ∞−∞ Y ( λ − µ ) e i √ µt √ µ [ ϕ ( x + t ) − ϕ ( x − t )] dµ dt = Z + ∞−∞ (Z λ e i √ µt √ µ dµ ) [ ϕ ( x + t ) − ϕ ( x − t )] dt for λ >
0. So, by setting u = √ µ we get( E λ ϕ ) ( x ) = Z + ∞−∞ dt π [ ϕ ( x + t ) − ϕ ( x − t )] Z λ e iut du. Now we have1 √ π Z λ e iut du = (cid:0) − e iλt (cid:1) F − ( Y ) ( t ) , so, using that F − ( Y ( v )) = F (1 − Y ( v )) ( t ) as well asequation (2.8), we get1 √ π Z λ e iut du = (cid:0) − e iλt (cid:1) r π (cid:18) δ ( t ) − iπ p . v . t (cid:19) which implies ( E λ ϕ ) ( x ) = − πi Z + ∞−∞ ϕ ( s ) s − x ds + 1 πi Z + ∞−∞ cos ( λ ( s − x )) s − x ϕ ( s ) ds where the integrals are understood in Cauchy’s principalvalue sense. (cid:3) Example 13.
Spectral family of the ( up to a sign ) one-dimensional Quantum Mechanics momentum operator S for a bounded particle on H = L [ − π, π ] with domain { ϕ ∈ L [ − π, π ] : ϕ ′ ∈ L [ − π, π ] , ϕ ( − π ) = ϕ ( π ) } As is well-known this is a self-adjoint operator with dis-crete spectrum σ ( S ) = Z whose eigenfunction system { ϕ n : n ∈ Z } , with ϕ n ( x ) = (2 π ) − / e − inx , are the solu-tions of the eigenvalue problem iϕ ′ = λϕ with ϕ ( − π ) = ϕ ( π ). So, for ϕ ∈ D ( S ) we have ϕ L = P n ∈ Z c n ϕ n with c n = h ϕ, ϕ n i = 12 π Z π − π ϕ ( x ) e inx dx for every n ∈ Z . Since σ ( S ) = σ d ( S ), recalling the defi-nition of the operator E λ for λ ∈ σ d ( S ), clearly we have( E λ ϕ ) ( x ) = lim ǫ → + Z + ∞−∞ Y ( λ + ǫ − µ ) ( δ ( µI − S ) ϕ ) ( x ) dµ for every λ ∈ R . So, the fact that E λ is a bounded operatoryields E λ ϕ = X n ∈ Z c n E λ ϕ n Using that δ ( µI − S ) e − inx = δ ( µ − n ) e − inx and that Y ( λ + 0 − n ) = Y ( λ − n ), we get( E λ ϕ ) ( x ) = X n ∈ Z c n √ π Y ( λ − n ) e − inx = X n ∈ Z , n ≤ [ λ ] e − inx √ π . Remark 14.
Since in the previous example S is boundedon H = L [ − π, π ], the delta operator δ ( λI − S ) shouldbe regarded as a continuous endomorphism as well. In thiscase δ ( λI − S ) ϕ = X n ∈ Z c n δ ( λ − n ) ϕ n . Example 15.
The one-dimensional Quantum Mechanicsposition operator on L ( R ) . This operator is defined on H = L ( R ) by ( Qϕ ) ( x ) = xϕ ( x ) for every x ∈ R . Clearly σ c ( Q ) = R and ϕ ∈ D ( Q ) if ( x x ϕ ( x )) ∈ L ( R ).Moreover, it is clear that { exp ( it ( λI − Q )) ϕ } ( x ) = e i ( λ − x ) t ϕ ( x ) . So we have( δ ( λI − Q ) ϕ ) ( x ) = δ ( λ − x ) ϕ ( x ) . Hence, in this case we can write { E λ ϕ } ( x ) = Z + ∞−∞ Y ( λ − µ ) δ ( µ − x ) ϕ ( x ) dµ Therefore, if λ = x we get { E λ ϕ } ( x ) = Y ( λ − x ) ϕ ( x ) . Example 16.
Explicit form of δ ( λI − M ) for the Her-mitian matrix of H = C M = . Proof.
In this case M = P J M P − with σ ( M ) = {− , } and J M = − − , P = − − Using (2.1) together with the fact that + ∞ Z −∞ e i ( λ − ρ ) t dt = 2 πδ ( λ − ρ ) , we get δ ( λI − M ) = 12 π P (cid:26)Z + ∞−∞ exp it ( λI − J M ) dt (cid:27) P − = P δ ( λ + 1) 0 00 δ ( λ + 1) 00 0 δ ( λ − P − Let us compute the spectral family and the projection op-erator onto the eigenspace ker ( M + I ). Clearly E λ = P Y ( λ + 1) 0 00 Y ( λ + 1) 00 0 Y ( λ − P − for every λ ∈ R . If λ = −
1, the orthogonal projection P λ onto ker ( I + M ) is P λ = 13 P P − = 13 − − − − − − since P λ = E λ − E λ − = E λ . (cid:3) Example 17.
Consider a compact self-adjoint operator K acting on a separable Hilbert space H which does notadmit the eigenvalue zero. Let { u i : i ∈ N } be a Hilbertbasis of H with its corresponding sequence of real eigen-values { λ i : i ∈ N } , where | λ i +1 | ≤ | λ i | for every i ∈ N .Let us compute the action of the operator ( λI − K ) − onany x ∈ H and the operator δ ( λI − T ) .Proof. If x ∈ H , we can write x = P ∞ i =1 h x, u i i u i . Since( λI − K ) − is a bounded operator whenever λ / ∈ σ ( K ),we have( λI − K ) − x = ∞ X i =1 h x, u i i Z + ∞−∞ λ − µ δ ( µI − K ) u i so we obtain the classic series( λI − K ) − x = ∞ X i =1 λ − µ i h x, u i i u i . For the solution of the equation ( I − zK ) x = y with z ∈ C we get the Schmidt series x = ( I − zK ) − y = ∞ X i =1 − zµ i h y, u i i u i whenever z − / ∈ σ ( T ). On the other hand, since δ ( λI − K )acts on H as a continuous endomorphism, equation δ ( λI − K ) x = ∞ P i =1 h x, u i i δ ( λ − µ i ) u i . holds for every x ∈ H . (cid:3) If T is an unbounded self-adjoint operator then D ( T ) = H and D ( T n ) becomes smaller as n grows. So, the follow-ing result, makes sense only if the operator T is bounded. Theorem 18.
In general, if T is a bounded self-adjointoperator, one has (2.9) δ ( λI − T ) = ∞ X n =0 ( − n δ ( n ) ( λ ) n ! T n which is the Taylor series of δ ( λI − T ) at λI .Proof. Developing the operator function exp ( itT ), whichis well-defined by the spectral theorem, we get δ ( λI − T ) = 12 π Z + ∞−∞ e − iλt ∞ X n =0 ( it ) n n ! T n dt, so that, formally interchanging the sum and the integral,we may write δ ( λI − T ) = 1 √ π ∞ X n =0 F { ( it ) n } ( λ ) n ! T n . Using the fact that
F { ( it ) n } ( λ ) = ( − n √ π δ ( n ) ( λ )for every n ∈ N , we obtain (2.9). (cid:3) The resolvent operator and δ ( λI − T )Recall that the spectrum σ ( T ) of a (densely defined)self-adjoint operator on a complex Hilbert space H is aclosed subset of C contained in R (see for instance [9, 3.2]).If z ∈ C \ σ ( T ), i. e., if z is a regular point of T , and R ( z, T ) = ( zI − T ) − denotes the resolvent operator of T at z (see [7, Definition8.2]), the function λ ( z − λ ) − is continuous on σ ( T ).The resolvent is well-defined over H , so it is a boundednormal operator. If z ∈ R \ σ ( T ) then R ( z, T ) is evenself-adjoint. From (1.1) it follows that R ( z, T ) = Z + ∞−∞ z − λ δ ( λI − T ) dλ which is the integral form of the resolvent of T . So, byconsidering the complex-valued function f ( λ ) = ( z − λ ) − with z ∈ C \ σ ( T ) and using the fact that F − (cid:18) λ − z (cid:19) ( t ) = √ π ie izt Y ( t )then, according to (2.5), for Im z > zI − T ) − = − i Z ∞ e izt U ( t ) . From here, it follows that R ( z, iT ) = i R ( iz, − T ) = (cid:0) L U − (cid:1) ( z )if Im z >
0, where L is the Laplace transform. Thisis the Hille-Yosida theorem which relates the resolventwith the one-parameter group of unitary transformations { U ( t ) : t ∈ R } generated by the self-adjoint operator T . If T is a bounded self-adjoint operator, γ is a closedJordan contour that encloses σ ( T ) and f ( z ) is holomor-phic inside the connected region surrounded by the path γ , the Dunford integral formula asserts that12 πi Z γ f ( z ) R ( z, T ) dz = f ( T ) . In [13] is pointed out that (2 πi ) − R ( z, T ) can be consid-ered as the indicatrix of a vector-valued distribution withvalues in L ( H ). Dunford integral formula is easily ob-tained by using the δ ( λI − T ) operator since, if we applythe Proposition 5 with g ( λ, µ ) = f ( z ( µ )) ( z ( µ ) − λ ) − ,where z ( µ ) = γ ( µ ) and 0 ≤ µ ≤
1, then Z γ f ( z ) R ( z, T ) dz = Z + ∞−∞ (cid:26)Z γ f ( z ) z − λ dz (cid:27) δ ( λI − T ) dλ = 2 πi Z + ∞−∞ f ( λ ) δ ( λI − T ) dλ = 2 πif ( T ) . Example 19.
Derivation of the orthogonal projection op-erator onto ker ( M + I ) of the Hermitian matrix M of theExample 16 by the resolvent technique. We must compute P λ = 12 πi Z | z +1 | =1 R ( z, M ) dz. Clearly, we have R ( z, M ) = 1 z − z − z − z − z − . Using that Z | z +1 | =1 { , z − } ( z + 1) ( z − dz = (cid:26) − πi , πi (cid:27) we reproduce the result we got earlier.4. The δ ( λI − T ) operator as a limit As µ ( λ ± iǫ − µ ) − is continuous, for self-adjoint T (( λ − iǫ ) I − T ) − − (( λ + iǫ ) I − T ) − = Z + ∞−∞ (cid:18) λ − iǫ − µ − λ + iǫ − µ (cid:19) δ ( µI − T ) dµ. If f ∈ D ( R ), Proposition 5 yields Z + ∞−∞ f ( λ )2 πi n (( λ − iǫ ) I − T ) − − (( λ + iǫ ) I − T ) − o dλ = 1 π Z + ∞−∞ (Z + ∞−∞ f ( λ ) ǫ ( λ − µ ) + ǫ dλ ) δ ( µI − T ) dµ. Since in the sense of distributions12 πi (cid:18) λ − iǫ − µ − λ + iǫ − µ (cid:19) → δ ( λ − µ )as ǫ → + , we have1 π Z R f ( λ ) ǫ ( λ − µ ) + ǫ dλ → Z R f ( λ ) δ ( λ − µ ) dλ as ǫ → + . Hence, if g n is defined by the left-hand side µ -parametric integral with ǫ = 1 /n , then g n → f point-wise on R . So, if f ∈ D ( R ) and T is bounded (hence with σ ( T ) compact), as can be easily checked { g n } ∞ n =1 is a uni-formly bounded sequence of continuous functions, withsup n ∈ N k g n k ∞ ≤ k f k ∞ , that converges pointwise on R to f . Thus, by [3, 10.2.8 Corollary] one has g n ( T ) → f ( T )in the strong operator topology, that is1 π Z + ∞−∞ (Z + ∞−∞ f ( λ ) ǫ ( λ − µ ) + ǫ dλ ) δ ( µI − T ) dµ → Z + ∞−∞ f ( µ ) δ ( µI − T ) dµ as ǫ → + in the strong operator topology of L ( H ). There-fore, if T is bounded and f ∈ D ( T ) then Z + ∞−∞ f ( λ )2 πi n (( λ − iǫ ) I − T ) − − (( λ + iǫ ) I − T ) − o dλ goes to f ( T ) as ǫ → + . This proves that for bounded T lim ǫ → + πi (( λ − iǫ ) I − T ) − − (( λ + iǫ ) I − T ) − coincides with δ ( λI − T ) as an L ( H )-valued distribution.5. Unitary equivalence of δ ( λI − T ) Theorem 20. If T is a self-adjoint operator defined onthe whole of H , there exist a finite measure µ on the Borelsets of the compact space σ ( T ) and a linear isometry U from L ( σ ( T ) , µ ) onto H such that U − δ ( λI − T ) U = δ ( λI − Q ) where ( Qϕ ) ( x ) = x ϕ ( x ) is the position operator.Proof. According to [5] there exist a finite measure µ onthe Borel sets of the compact space σ ( T ) and a linearisometry U from L ( σ ( T ) , µ ) onto H such that (cid:0) U − T U (cid:1) ϕ = Qϕ for every ϕ ∈ L ( σ ( T ) , µ ). So, since U − T U is a self-adjoint operator on L ( σ ( T ) , µ ), we have U − δ ( λI − T ) U = δ (cid:0) λI − U − T U (cid:1) = δ ( λI − Q )as stated. (cid:3) Remark 21.
For such linear isometry U the equation (cid:0) U − δ ( λI − T ) U ϕ (cid:1) ( x ) = δ ( λ − x ) ϕ ( x ) holds for every ϕ ∈ L ( σ ( T ) , µ ) . Commutation relations
Let S and T be two self-adjoint operators defined on thewhole of H for which equations [ S, [ S, T ]] = [ T, [ S, T ]] = hold. In this case[ − itS, [ − itS, − isT ]] = it s [ S, [ S, T ]] = and the Baker-Campbell-Hausdorff formula yields δ ( λI − S ) δ ( µI − T ) =1(2 π ) Z Z R e i ( tλ + sµ ) e − itS e − isT dt ds =1(2 π ) Z Z R e i ( tλ + sµ ) e − st [ S,T ] e − i ( tS + sT ) dt ds. Likewise, since [ T, [ T, S ]] = [ S, [ T, S ]] = one has δ ( µI − T ) δ ( λI − S ) =1(2 π ) Z Z R e i ( tλ + sµ ) e − st [ T,S ] e − i ( tT + sS ) dt ds = 1(2 π ) Z Z R e i ( tλ + sµ ) e st [ S,T ] e − i ( tS + sT ) dt ds. So, using thatexp (cid:18) ist i [ S, T ] (cid:19) − (cid:18) − ist i [ S, T ] (cid:19) = 2 i sin (cid:18) ist S, T ] (cid:19) we have [ δ ( λI − S ) , δ ( µI − T )] = i π Z Z R sin (cid:18) ist S, T ] (cid:19) e i ( tλ + sµ ) e − i ( tS + sT ) dt ds. For position Q and momentum P of a one-dimensionalparticle, one has H = L ( R ) and [ Q, P ] = i ~ I . Therefore[ Q, [ Q, P ]] = [ P, [ Q, P ]] = and[ δ ( λI − Q ) , δ ( µI − P )] = i π Z Z R sin (cid:18) − st ~ (cid:19) e i ( tλ + sµ ) e − i ( tQ + sP ) dt ds. According to Theorem 8, if [ δ ( λI − S ) , δ ( µI − T )] actson f ( λ ) = λ , formally we have Z + ∞−∞ λ [ δ ( λI − S ) , δ ( µI − T )] dλ = i π Z Z R sin (cid:18) ist S, T ] (cid:19)(cid:26)Z + ∞−∞ λe itλ dλ (cid:27) e isµ e − i ( tS + sT ) dt ds. So, using the distributional equality(6.1) Z + ∞−∞ λe itλ dλ = 2 πi δ ′ ( t )and integrating by parts, it follows that Z + ∞−∞ λ [ δ ( λI − S ) , δ ( µI − T )] dλ = − i π [ S, T ] Z + ∞−∞ se is ( µI − T ) ds. Observe that a second application of equation (6.1) and asecond integration by parts yield Z + ∞−∞ λµ [ δ ( λI − S ) , δ ( µI − T )] dλ dµ = − i π [ S, T ] Z + ∞−∞ se − isT (cid:26)Z + ∞−∞ µe isµ dµ (cid:27) ds =[ S, T ] Z + ∞−∞ δ ( s ) { − isT } e − isT ds = [ S, T ]as expected.7.
A remark on the Stone formula
Let T be a self-adjoint operator densely defined on aHilbert space H . If A is a Borel set in σ ( T ), defining(7.1) E ( A ) := Z + ∞−∞ χ A ( λ ) dE λ where χ A stands for the characteristic function of A (whichis a bounded Borel function), then E is an L ( H )-valuedfinitely additive and pointwise countably additive measure(i.e., countable additivity under the strong operator topol-ogy of L ( H )) on the σ -algebra A of Borel subsets of σ ( T ).So, if the characteristic function χ A of A with respect to R is continuous on σ p ( T ) then E ( A ) = Z + ∞−∞ χ A ( λ ) δ ( λI − T ) dλ. For −∞ < a < b < ∞ and ǫ >
0, we have Z ba (cid:26)Z + ∞−∞ (cid:18) λ − iǫ − µ − λ + iǫ − µ (cid:19) δ ( µI − T ) dµ (cid:27) dλ = Z + ∞−∞ (Z ba iǫ dλ ( λ − µ ) + ǫ ) δ ( µI − T ) dµ =2 i Z R (cid:26) arg tan (cid:18) b − µǫ (cid:19) − arg tan (cid:18) a − µǫ (cid:19)(cid:27) δ ( µI − T ) dµ. If the limit as ǫ → + the bracketed function is equal to 0if µ ∈ R \ [ a, b ], equal to π if a < µ < b and equal to π/ µ ∈ { a, b } . So, if a, b / ∈ σ p ( T ) so that χ ( a,b ) and χ [ a,b ] both belong to B p ( R ), setting g n ( µ ) := 1 π Z ba in − dλ ( λ − µ ) + n − for each n ∈ N and f ( µ ) := χ ( a,b ) ( µ ) + χ [ a,b ] ( µ ) , then g n ( µ ) → f ( µ ) for every µ ∈ R and sup n ∈ N k g n k ∞ ≤ g n ( T ) x → f ( T ) x for every x ∈ D ( T ). In other wordslim ǫ → + πi Z ba (cid:16) ( λ − iǫ − T ) − − ( λ + iǫ − T ) − (cid:17) dλ = 12 Z + ∞−∞ (cid:16) χ ( a,b ) + χ [ a,b ] (cid:17) δ ( µI − T ) dµ, holds pointwise on the domain D ( T ) of T . Hence, byvirtue of (7.1) we getlim ǫ → + πi Z ba (cid:16) ( λ − iǫ − T ) − − ( λ + iǫ − T ) − (cid:17) dλ =12 E (( a, b )) + 12 E ([ a, b ]) = E ( a, b ) + 12 E ( a ) + 12 E ( b )which is Stone’s formula. References [1] Dirac, P. A. M.,
The Physical Interpretation of the QuantumDynamics , Proc. Royal Soc. London (1927), 621–641.[2] Dirac, P. A. M.,
The Principles of Quantum Mechanics , 4thEdition, Clarendon Press, Oxford, 1988.[3] Dunford, N. and Schwartz, J. T.,
Linear Operators. Part II : Spectral Theory , John Wiley, New York, 1988.[4] Griffiths, D. and Walborn, S.,
Dirac deltas and discontinuousfunctions , Am. J. Phys. (1999), 446–446.[5] Halmos, P., What does the spectral theorem say?
Amer. Math.Monthly (1963), 241–247.[6] Horv´ath, J., Topological Vector Spaces and Distributions , Vol.I, Addison-Wesley, Reading, Massachusetts, 1966.[7] Moretti, V.,
Spectral Theory and Quantum Mechanics , 2nd Edi-tion, Unitext , Springer, 2013.[8] Riesz, F., Nagy, B. Sz.-,
Functional Analysis , Dover Publica-tions, Mineola, 1990.[9] Schm¨udgen, K.,
Unbounded Self-adjoint Operators on HilbertSpace , Springer, Dordrecht, 2012.[10] Schwartz, L.,
Th´eorie des distributions `a valeurs vectorielles.I , Anal. Inst. Fourier (1957), 1–141.[11] Schwartz, L., Th´eorie des distributions `a valeurs vectorielles.II , Anal. Inst. Fourier (1958), 1–209.[12] Schwartz, L., Th´eorie des Distributions , Hermann, Paris, 1966.[13] Tillmann, H. G.,
Vector-valued distributions and the spectraltheorem for self-adjoint operators in Hilbert space , Bull. Amer.Math. Soc. (1963), 67–71. Centro de Investigaci´on Operativa, UniversidadMiguel Hern´andez, E-03202 Elche, Spain
Email address ::